Avoiding order reduction with explicit Runge-Kutta exponential methods in nonlinear initial boundary value problems

TL;DR

This paper presents a technique to maintain the classical order of explicit exponential Runge-Kutta methods when solving nonlinear reaction-diffusion problems with boundary conditions, reducing computational cost and handling time-dependent boundaries effectively.

Contribution

The paper introduces a boundary information addition technique that preserves the order of explicit exponential Runge-Kutta methods in reaction-diffusion problems with boundary conditions.

Findings

The method effectively recovers classical order in reaction-diffusion problems.

It handles time-dependent boundary conditions directly.

Reduces computational cost compared to high stiff order methods.

Abstract

In this paper a technique is given to recover the classical order of the method when explicit exponential Runge-Kutta methods integrate reaction-diffusion problems. Although methods of high stiff order for problems with vanishing boundary conditions can be constructed, that may imply increasing the number of stages and therefore, the computational cost seems bigger than the technique which is suggested here, which just adds some terms with information on the boundaries. Moreover, time-dependent boundary conditions are directly tackled here.